Volume 16, issue 6 (2016)

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Rigidification of higher categorical structures

Giovanni Caviglia and Geoffroy Horel

Algebraic & Geometric Topology 16 (2016) 3533–3562
Abstract

Given a limit sketch in which the cones have a finite connected base, we show that a model structure of “up to homotopy” models for this limit sketch in a suitable model category can be transferred to a Quillen-equivalent model structure on the category of strict models. As a corollary of our general result, we obtain a rigidification theorem which asserts in particular that any ${\Theta }_{n}$–space in the sense of Rezk is levelwise equivalent to one that satisfies the Segal conditions on the nose. There are similar results for dendroidal spaces and $n$–fold Segal spaces.

Keywords
internal operads, internal n-categories, limit sketches, model categories
Mathematical Subject Classification 2010
Primary: 18C30, 18D35, 55U35
Secondary: 18D05, 18D50
Publication
Received: 13 November 2015
Revised: 16 May 2016
Accepted: 16 May 2016
Published: 15 December 2016
Authors
 Giovanni Caviglia Institute for Mathematics, Astrophysics, and Particle Physics Radboud University Nijmegen Heyendaalseweg 135 6525 AJ Nijmegen Netherlands http://www.math.ru.nl/~gcaviglia/ Geoffroy Horel Max Planck Institute for Mathematics Vivatsgasse 7 D-53111 Bonn Germany http://geoffroy.horel.org/